1. Field of the Invention
The present invention generally relates to a system and method for load balancing of multi-phase electric distribution feeders and, in particular, to an algorithm usable to determine optimal tap changes for phase swapping.
2. Description of the Related Art
Phase balancing aims to reduce three phase load unbalance, to avoid severe voltage drops in electric power feeder circuits. The majority of electric power distribution systems utilize feeders that carry three-phase alternating power. It is desirable for electric utilities to provide approximately equal loads on each phase. A problem arises in that, even for initially balanced loads, over time as the loads change a loads unbalance will arise. Significant load variation on feeder phases can arise during a single day.
There are two major phase balancing methods: feeder reconfiguration at a system level and phase swapping at the feeder level. See, J. Zhu, G. Bilbro, M. Chow, Phase Balancing using Simulated Annealing, IEEE Power Sys. Trans., Vol. 14, No. 4, pp. 1508-1513, November 1999. In electric power literature, phase swapping is less studied than feeder reconfiguration.
In a three-phase system, phase unbalance limits an amount of power transferred on a feeder, since one phase of an unbalanced feeder will reach its maximum carrying capacity while the other phases are underutilized. Such poor utilization of a feeder in a power distribution network can result in unnecessary outages, unnecessary feeder expansion, and unnecessary system upgrades, resulting in decreased reliability and increased utility costs. As the highest loaded phase nears maximum ampacity, phase unbalance can lead to preventive breaker/relay tripping and feeder shutdown, restoration of which will increase electric utility operating costs.
Electric utility crews periodically rebalance feeders, typically during maintenance and restoration. For example, a suburban northeast U.S. utility will rebalance feeders once the unbalance exceeds 15%. Generally, ten to fifteen minutes are needed to perform load switching. A complete load switch can take an hour, excluding travel time to a location where the load switch must be performed. Completion of a load switch by a crew of two employees can cost several hundred dollars. Additional preparatory work, such as scheduling, can bring the total cost of rebalancing to several thousand dollars for a single tap change.
Tap changes generally occur when a new customer is to be connected or the phase balance on an existing feeder becomes significantly unbalanced. Rebalancing a feeder is not a permanent solution, since a re-balanced feeder can readily drift into unbalance over time. The three factors considered in making a decision to rebalance a feeder are typically monetary cost of making the tap changes, expected increase in feeder balance, i.e., energy savings, and duration of customer power interruption.
Similar problems of effective phase balancing may arise in limited electric power systems, such as electric power systems provided in a tactical battlefield environment, often due to insufficient operator training and experience. See M. N. Gaffney, Intelligent Power Management: Improving Power Distribution in the Field. Phase balancing methods have been proposed. See, U.S. Pat. No. 5,604,385 to David, U.S. Pat. No. 6,018,203 to David, and U.S. Pat. No. 7,242,110 to Matusmoto, et al. Variables in the phase balancing problem are identification of phases connected to each load, with a goal to minimize a degree of feeder unbalance. Algorithms have been proposed to solve phase balancing problem. See, J. Zhu, et al. (IEEE Power Sys Trans, November 1998), which proposes a mixed-integer algorithm. However, the mixed-integer algorithm has a drawback that the objective functions can only be linear. As mentioned above, J. Zhu, et al. (IEEE Power Sys Trans, November 1999), propose expanding nonlinear objective functions by introduction of simulated annealing.
In 2000, a genetic algorithm was proposed. See, Chen, T. H., et al., Optimal Phase Arrangement of Distribution Transformers Connected to a Primary Feeder for System Unbalance Improvement and Loss Reduction Using a Genetic Algorithm, IEEE Power Sys. Trans, Vol. 15, No. 3, August 2000, pp 994-1000. Also see, Gandomkar, M., Phase Balancing Using Genetic Algorithm, 39th Int'l. Univ. Power Engineering Conf., September 2004, pp. 377-379. A heuristic greedy algorithm has also been proposed. See, Lin, Chia-Hung, et al., Heuristic Rule-Based Phase Balancing of Distribution Systems by Considering Customer Load Patterns, IEEE Power Sys. Trans., Vol. 20, No. 2, May 2005, pp 709-716. An immune algorithm has also been proposed. See, Huang, M-Y., et al., Three-phase Balancing of Distribution Feeders Using Immune Algorithm, IET Gen., Trans. and Dist., August 2007, pp. 383-392. These heuristic algorithms can get near-optimal solution quickly, but fail to guarantee optimal solutions.
The combinatorial optimization problems have not produced any known efficient algorithms capable of always producing optimal solutions. For those problems that computer scientists have been shown to be NP-Complete (NPC), there is convincing evidence that no correct, efficient algorithms can exist. An efficient algorithm for any one of the hundreds of known NPC problems would imply efficient algorithms for all of them, implying that all are equally hard to compute.
A phase balancing problem exists that is the equivalent to integer partitioning, a well-known NPC problem. An efficient algorithm for phase balancing which always produced optimal solutions would imply efficient algorithms for all problems in NP, which computer scientists consider extremely unlikely. However, heuristic algorithms that produce near optimal solutions with reasonable efficiency are possible. See, Skiena, S., The Algorithm Design Manual, 2d Ed., Springer, 2008.
Therefore, a dynamic programming algorithm is introduced to obtain an optimal solution for phase balancing problem in a reasonable running time, and to balance both the entire feeder and each section along the feeder, thereby avoiding a shortcoming of conventional systems in which the three phase current is balanced at the beginning of the feeders, but is not balanced at other positions of the feeders. See, Wang, K., Skiena, S., and Robertazzi, T. G, Phase Balancing Algorithms, Elec. Power Sys. Research, Vol. 96, March 2013, pp. 218-224.
Accordingly, an apparatus and method utilizing a dynamic programming algorithm are provided that solves phase balancing problems along each part of the feeder that conventional systems fail to address.